the person who has the dog now also has a child and 2 cats
Fostered with them by the rescue, or a home-to-home adoption where the puppy is leaving its original family to come straight to her?
Because in the second case, why are they giving up the puppy?
"a child" is not her 5-month-old baby.
She has a massive amount on her plate already, and this particular dog is almost bound to be the wrong decision - it won't be her old dog. It will be a young, boundary-pushing puppy who needs time and training.
We've just rehomed a 2 year old calm and lovely dog, and my DC are older, but not old enough to be left home alone while someone walks the dog. On days DH isn't around for whatever reason, this can be a battleground or a logistical nightmare. If walking a puppy with a pram isn't much fun, chasing a poorly trained dog in one direction when the toddler with a tantrum refuses to walk further or runs in the opposite direction will be hell.
Walks to the park are great - but the pup will need to be tied up outside the play area if she wants to push the toddler on the swings.
Inside the house, the dog will need to be gated away from the baby most of the time unless she can give her undivided attention to both baby and dog at the same time.
Extra cleaning, extra expenses - it's just a massively bad idea all round.
I recommend a nice friendly cat.